Struggling with finding matrices such as AC = AD when C is not equal D and A has no elements equal to zero.

Question

I tried to find matrices that will satisfy the condition of AC = AD when C is not equal D and A has no elements equal to zero, however I could find any right option since when it comes to system of equations it becomes tough since non of the A entries are 0 and I can't find the right elements of C and D. Please help or describe a functioning algorithm of finding those matrices

Answer 1

Since $A(C-D)=0$, you just need to find an $A$ with nontrivial null space (maybe that’ll be more clear if you think about $C-D$ as a matrix with vectors $v_1,v_2,..,v_n$ as columns).

As an example how one might do that. Consider the $2$ by $2$ matrix,

$$A=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$$

Obviously the first row minus the second row gives $\mathbf{0}$, so $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$ is in the null space. So let $C$ and $D$ be any two matrices such that:

$$C-D=\begin{pmatrix} 1 \\ -1 \end{pmatrix}$$

Or,

$$C-D=\begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}$$

Or,

$$C-D=\begin{pmatrix} 1 & 1 & 1 \\ -1 & -1 &-1 \end{pmatrix}$$

Etc...